Electronically scanned RF-to-bits beam aperture arrays using 2-D IIR spatially bandpass digital filters

A beamforming system based on two-dimensional (2-D) spatially bandpass infinite impulse response (IIR) plane wave filtering is presented in a multi-dimensional signal processing perspective and the implementation details are discussed. Real-time implementation of such beamforming systems requires modeling of computational electromagnetics for the antennas, radio frequency (RF) analog design aspects for low-noise amplifiers (LNAs), mixed-signal aspects for signal quantization and sampling and finally, digital architectures for the spatially bandpass plane wave filters proposed in Joshi et al. (IEEE Trans Very Large Scale Integr Syst 20(12):2241–2254, 2012). Multi-dimensional spatio-temporal spectral properties of down-converted RF plane wave signals are reviewed and derivation of the spatially bandpass filter transfer function is presented. An example of a wideband antipodal Vivaldi antenna is simulated at 1 GHz. Potential RF receiver chains are identified including a design of a tunable combline microstrip bandpass filter with tuning range 0.8–1.1 GHz. The 1st-order sensitivity analysis of the beam filter 2-D $\mathbf z $ -domain transfer function shows that for a 12-bits of fixed-point precision, the maximum percentage error in the 2-D magnitude frequency response due to quantization is as low as $0.3\,\%$ . Monte-Carlo simulations are used to study the effect of quantization on the bit error rate (BER) performance of the beamforming system. 5-bit analog to digital converter (ADC) precision with 8-bit internal arithmetic precision provides a gain of approximately 16 dB for a BER of $10^{-3}$ with respect to the no beamforming case. ASIC Synthesis results of the beam filter in 45 nm CMOS verifies a real time operating frequency of 429 MHz.     

Polyphase Conditions and Structures for 2-D Quincunx FIR Filter Banks Having Quadrantal or Diagonal Symmetries

In this brief, we derive conditions on the polyphase matrix of 2-D finite-impulse response (FIR) quincunx filter banks, for the filters in the filter bank to have quadrantal or diagonal symmetry. These conditions provide a framework for synthesizing polyphase structures which structurally enforce the symmetry. This is demonstrated by constructing examples of small parameterized matrix structures which satisfy the above conditions, thus giving perfect reconstruction FIR quincunx filter banks with quadrantal or diagonally symmetric short-kernel (i.e., short-support) filters. It is also shown that cascades of the above constructed small structures can be used to construct filters of higher order.     

Fast Implementation of 3-D Digital Filters Via Systolic Array Processors

In this paper a fast implementation architecture of three-dimensional (3-D) FIR or IIR digital filters via systolic VLSI array processors is described. The modular structure presented is comprised of similar processing elements in a linear cascade configuration with local interconnections. High speed throughput rates are attained due to high concurrency, which is achieved by exploiting both pipelining and parallelism. The considered 3-D FIR and IIR filters may be used for the processing of reconstructed 3-D images and in medical imaging applications.     

Design of highly selective two-dimensional recursive fan filters byrelaxing symmetry constraints

Shows for the design of quadrantally symmetric 2-D fan filters that it is unnecessarily restrictive to prescribe exact quadrantal symmetry, which requires that the denominator of the Z-transform transfer function be product-separable. Superior approximately symmetric fan filter designs can be achieved using nonseparable denominators     

Unified theory of symmetry for two-dimensional complex polynomials using delta discrete-time operator

The complexity in the design and implementation of 2-D filters can be reduced considerably if the symmetries that might be present in the frequency responses of these filters are utilized. As the delta operator (??-domain) formulation of digital filters offers better numerical accuracy and lower coefficient sensitivity in narrow-band filter designs when compared to the traditional shift-operator formulation, it is desirable to have efficient design and implementation techniques in ??-domain which utilize the various symmetries in the filter specifications. Furthermore, with the delta operator formulation, the discrete-time systems and results converge to their continuous-time counterparts as the sampling periods tend to zero. So a unifying theory can be established for both discrete- and continuous-time systems using the delta operator approach. With these motivations, we comprehensively establish the unifying symmetry theory for delta-operator formulated discrete-time complex-coefficient 2-D polynomials and functions, arising out of the many types of symmetries in their magnitude responses. The derived symmetry results merge with the s-domain results when the sampling periods tend to zero, and are more general than the real-coefficient results presented earlier. An example is provided to illustrate the use of the symmetry constraints in the design of a 2-D IIR filter with complex coefficients. For the narrow-band filter in the example, it can be seen that the ??-domain transfer function possesses better sensitivity to coefficient rounding than the z-domain counterpart.     

Design of 3-D Noncausal Filters with Small Roundoff Noise and No Overflow Oscillations

The contribution of this paper consists of two individual parts. First, an invertible mapping technique is presented for 3-D digital system design, and it is applied to approximate 3-D noncausal filters in the spatial domain. Secondly, an algorithm is proposed for obtaining a structure for 3-D IIR filters with small roundoff noise and no overflow oscillations. The design of noncausal filters can be carried out by three steps: 1), a given noncausal impulse response is transformed into the first octant using the proposed 3-D invertible mapping technique; 2), the transformed impulse response in the first octant is approximated by balanced model reduction of 3-D separable denominator systems;3), the resultant 3-D IIR filter is transformed back to the original coordinates.     

A new class of two-channel biorthogonal filter banks and waveletbases

Proposes a novel framework for a new class of two-channel biorthogonal filter banks. The framework covers two useful subclasses: i) causal stable IIR filter banks. ii) linear phase FIR filter banks. There exists a very efficient structurally perfect reconstruction implementation for such a class. Filter banks of high frequency selectivity can be achieved by using the proposed framework with low complexity. The properties of such a class are discussed in detail. The design of the analysis/synthesis systems reduces to the design of a single transfer function. Very simple design methods are given both for FIR and IIR cases. Zeros of arbitrary multiplicity at aliasing frequency can be easily imposed, for the purpose of generating wavelets with regularity property. In the IIR case, two new classes of IIR maximally flat filters different from Butterworth filters are introduced. The filter coefficients are given in closed form. The wavelet bases corresponding to the biorthogonal systems are generated. the authors also provide a novel mapping of the proposed 1-D framework into 2-D. The mapping preserves the following: i) perfect reconstruction; ii) stability in the IIR case; iii) linear phase in the FIR case; iv) zeros at aliasing frequency; v) frequency characteristic of the filters     

Optimization of two-dimensional IIR filters with nonseparable and separable denominator

We present algorithms for the optimization of two-dimensional (2-D) infinite impulse response (IIR) filters with separable or nonseparable denominator, for least squares or Chebyshev criteria. The algorithms are iterative, and each iteration consists of solving a semidefinite programming problem. For least squares designs, we adapt the Gauss-Newton idea, which outcomes to a convex approximation of the optimization criterion. For Chebyshev designs, we adapt the iterative reweighted least squares (IRLS) algorithm; in each iteration, a least squares Gauss-Newton step is performed, while the weights are changed as in the basic IRLS algorithm. The stability of the 2-D IIR filters is ensured by keeping the denominator inside convex stability domains, which are defined by linear matrix inequalities. For the 2-D (nonseparable) case, this is a new contribution, based on the parameterization of 2-D polynomials that are positive on the unit bicircle. In the experimental section, 2-D IIR filters with separable and nonseparable denominators are designed and compared. We show that each type may be better than the other, depending on the design specification. We also give an example of filter that is clearly better than a recent very good design.     

Frequency domain design of FIR and IIR Laplacian of Gaussian filters for edge detection

This work addresses the design of LoG filters in the frequency domain within a structure formed by the cascade of quasi-Gaussian and discrete Laplacian filters. The main feature of such a structure is that it requires half the number of convolutions of the classical structure in which the LoG transfer function is expressed as the sum of two separable transfer functions of 1-D Gaussian and LoG type. Such a perspective allows one to rephrase the design of IIR and FIR filters for edge detection as a frequency domain approximation problem solvable by standard digital filter design tools. The zero-phase IIR solutions have a good performance at low orders and approximation errors practically independent of the aperture parameter. The characteristics of the nearly linear-phase IIR filters solving the problem suggest the consideration of linear-phase FIR filters with zeros constrained on the unit circle. The use of such filters leads to remarkable computational savings with respect to the filters designed by impulse response sampling. The agreement between the edge values obtained by the filters designed according to the scheme proposed in this work and those obtained by standard techniques is very good.Work carried out with the financial support of the C.N.R.-Progetto Finalizzato Robotica, contract no. 91.01942.PF67.     

一种数字全通滤波器的设计方法研究

研究了一种数字全通滤波器的设计方法.对于一个平稳的全通滤波器,其分母多项式一定具有最小相位.该方法是基于最小相位滤波器的复倒谱系数和其群迟延函数以及其系统函数之间的关系,通过一个非线性的递归方程求解分母多项式的系数.由全通滤波器的特性已知,分母系数可以完全决定全通滤波器的传递函数.仿真结果表明这种方法能够使所设计滤波器的群延迟特性在整个频率范围内以近似理想的群延迟特性存在.并结合实现提出了一种用FIR逼近IIR的方法.     

Design of 2-D FIR narrow transition band filters by double transformations using GA approach

In this investigation, subfilters are cascaded in the design of a 2-D narrow transition band FIR digital filter with double transformations, a transformation from wide transition band subfilter into 1-D narrow transition band filter and a McClellan transformation from 1-D filter into 2-D filter. The traditional method for designing a 2-D FIR digital filter with a narrow transition band yields very high orders. The difficulty of the design and implementation will increase with orders exponentially. Numerous identical low-order subfilters are cascaded together to simplify the design of a high-order 2-D filter compared to traditional design method. A powerful genetic algorithm (GA) is presented to determine the best coefficients of the McClellan transformation. It can be used to design any contours of arbitrary shape for mapping 1-D to 2-D FIR filters very effectively. A generalized McClellan transformation is presented, and can be used to design 2-D complex FIR filters. Various numerical design examples are presented to demonstrate the usefulness and effectiveness of the presented approach.

FIR, Allpass, and IIR Variable Fractional Delay Digital Filter Design

This paper presents two-step design methodologies and performance analyses of finite-impulse response (FIR), allpass, and infinite-impulse response (IIR) variable fractional delay (VFD) digital filters. In the first step, a set of fractional delay (FD) filters are designed. In the second step, these FD filter coefficients are approximated by polynomial functions of FD. The FIR FD filter design problem is formulated in the peak-constrained weighted least-squares (PCWLS) sense and solved by the projected least-squares (PLS) algorithm. For the allpass and IIR FD filters, the design problem is nonconvex and a global solution is difficult to obtain. The allpass FD filters are directly designed as a linearly constrained quadratic programming problem and solved using the PLS algorithm. For IIR FD filters, the fixed denominator is obtained by model reduction of a time-domain average FIR filter. The remaining numerators of the IIR FD filters are designed by solving linear equations derived from the orthogonality principle. Analyses on the relative performances indicate that the IIR VFD filter with a low-order fixed denominator offers a combination of the following desirable properties including small number of denominator coefficients, lowest group delay, easily achievable stable design, avoidance of transients due to nonvariable denominator coefficients, and good overall magnitude and group delay performances especially for high passband cutoff frequency ( ges 0.9pi) . Filter examples covering three adjacent ranges of wideband cutoff frequencies [0.95, 0.925, 0.9], [0.875, 0.85, 0.825], and [0.8, 0.775, 0.75] are given to illustrate the design methodologies and the relative performances of the proposed methods.     

Generating 2-D FIR filter functions by Christoffel–Darboux formula for Chebyshev polynomials of the second kind

Global Christoffel–Darboux formula for different polynomials has already been used for the filter design. Here, this formula for orthonormal Chebyshev polynomials of the second kind and for two independent variables is applied in generating novel class linear-phase two-dimensional (2-D) finite impulse response (FIR) digital filter functions. In this way, 2-D filters with some specific features including economy, phase linearity, symmetry and selectivity are designed. Representative examples of the 2-D FIR digital filters of a new class obtained by the proposed approximation technique are given. A filter generated by the proposed approach is compared with the corresponding one generated by the procedure from literature.     

Design of stable, causal, perfect reconstruction, IIR uniform DFTfilter banks

Design procedures for stable, causal and perfect reconstruction IIR parallel uniform DFT filter banks (DFT FBs) are presented. In particular a family of IIR prototype filters is a good candidate for DFT FB, where a tradeoff between frequency selectivity and numerical properties (as measured by the Weyl-Heisenberg frames theory) could be made. Some realizations exhibiting a simple and a massively parallel and modular processing structure making a VLSI implementation very suitable are shown. In addition, some multipliers in the filters (both the analysis and synthesis) could be made; powers or sum of powers of 2, in particular for feedback loops, resulting in a good sensitivity behavior. For these reasons as well as for the use of low order IIR filters (as compared with conventional FIR filters), the overall digital filter bank structure is efficient for high data rate applications. Some design examples are provided     

A new approach to the design of biorthogonal triplet half-band filter banks using generalized half-band polynomials

This paper presents a novel approach to design a class of biorthogonal triplet half-band filter banks based on the generalized half-band polynomials. The filter banks are designed with the help of three-step lifting scheme (using three kernels). The generalized half-band polynomial is used to construct these three kernels by imposing the number of zeros at \(z=-1\) . The maximum number of zeros imposed for the three kernels is half of the order of half-band polynomial ( \(K/2\) for \(K\) order polynomial). The three kernels give a set of constraints on the coefficients of half-band polynomial by imposing the zeros. In addition to structural perfect reconstruction and linear phase, the proposed filter banks provide better frequency selectivity, more similarity between analysis and synthesis filters (measure of near-orthogonality), and good time–frequency localization. The proposed technique offers more flexibility in the design of filters using two degrees of freedom. Some examples have been presented to illustrate the method.     

VLSI architectures for high-speed and flexible two-dimensionaldigital filters

Flexible VLSI architectures for high-speed 2-D finite-impulse-response (FIR) and infinite-impulse-response (IIR) digital filters are described. Cyclical parallel processing structures for 2-D FIR and IIR digital filtering are derived from the employment of storage elements. The hardware architectures that realize the parallel processing structures are developed. The resulting architectures, which are mainly constructed of three types of standard cells, exhibit a high degree of modularity and regularity, and thus a high suitability for VLSI implementation. The architectures can process 2-D data arrays of arbitrary dimensions in real time or near real time and have higher hardware efficiency and lower implementation cost than the direct-form realization     

Decomposition of FIR digital filters for realization via thecascade connection of subfilters

A method is presented for decomposing even-order linear-phase FIR filters with distinct roots into the cascade connection of second-and fourth-order subfilters. The technique consists of of finding roots of the z-domain filter transfer function by searching a finite region in the complex z plane. Due to symmetry in the impulse response, only the perimeter (the real axis and boundary) and interior of the upper half of the unit circle need to be searched for real and complex values of roots from which the impulse response coefficients of the corresponding subfilters can be directly determined. The root-finding algorithm tests for existence of a root at each interval in a finite grid and then utilizes the Newton-Raphson method to refine the final estimate of each root value. In the two-dimensional (2-D) search, the Cauchy-Riemann relations are exploited to reduce computations and speed convergence. This method has been tested on FIR filters with orders ranging to over 120 and has proven effective in decomposing filters to the cascade realization with identical frequency response characteristics. An example is presented that illustrates the use of this technique     

Two-dimensional digital filtering

The problems of designing and implementing LSI systems for the processing of 2-D digital data, such as images or geophone arrays, are reviewed and discussed. This discussion encompasses both FIR and IIR digital filters and with respect to the latter, the issues of stability testing and filter stabilization are also considered. Techniques are also presented whereby such filtering can be accomplished using either 1 or 2-D LSI systems.